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<br> <h1> Link to the source code for this example: </h1>
<a href="https://github.com/libMesh/libmesh/tree/master/examples/adjoints/adjoints_ex1" target="_blank">Open adjoints_ex1 in new tab.</a>
<a name="output"></a> 
<br><br><br> <h1> The console output of the program: </h1> 
<pre>
***************************************************************
* Running Example adjoints_ex1:
*   example-opt  
***************************************************************
 
Started /Users/petejw/projects/libmesh_git/build/examples/adjoints/adjoints_ex1/.libs/example-opt
Reading in and building the mesh
Building system
Initializing systems
 Mesh Information:
  elem_dimensions()={2}
  spatial_dimension()=2
  n_nodes()=833
    n_local_nodes()=833
  n_elem()=255
    n_local_elem()=255
    n_active_elem()=192
  n_subdomains()=1
  n_partitions()=1
  n_processors()=1
  n_threads()=1
  processor_id()=0

 EquationSystems
  n_systems()=1
   System #0, "LaplaceSystem"
    Type "Implicit"
    Variables="T" 
    Finite Element Types="LAGRANGE", "JACOBI_20_00" 
    Infinite Element Mapping="CARTESIAN" 
    Approximation Orders="SECOND", "THIRD" 
    n_dofs()=833
    n_local_dofs()=833
    n_constrained_dofs()=0
    n_local_constrained_dofs()=0
    n_vectors()=1
    n_matrices()=1
    DofMap Sparsity
      Average  On-Processor Bandwidth <= 15.06
      Average Off-Processor Bandwidth <= 0
      Maximum  On-Processor Bandwidth <= 25
      Maximum Off-Processor Bandwidth <= 0
    DofMap Constraints
      Number of DoF Constraints = 0

Assembling the System
Nonlinear Residual: 5.00981e+11
Linear solve starting, tolerance 1e-12
Linear solve finished, step 60, residual 6.55266e-12
Trying full Newton step
  Current Residual: 46.9133
  Nonlinear solver converged, step 0, residual reduction 9.3643e-11 < 1e-07
  Nonlinear solver relative step size inf > 1e-07
Adaptive step 0, we have 192 active elements and 833 active dofs.
Postprocessing: 
The relative error in QoI 0 is 0.00012462746009579801
The relative error in QoI 1 is 0.00014255923631530034

Using Adjoint Residual Error Estimator with Patch Recovery Weights
Refined mesh to 222 active elements and 945 active dofs.
Assembling the System
Nonlinear Residual: 204936.50034704173
Linear solve starting, tolerance 9.9999999999999998e-13
Linear solve finished, step 63, residual 1.025550879191738e-14
Trying full Newton step
  Current Residual: 3.3527977950047243e-05
  Nonlinear solver converged, step 0, residual reduction 1.6360178832599657e-10 < 9.9999999999999995e-08
  Nonlinear solver relative step size 0.00097606015940252867 > 9.9999999999999995e-08
Adaptive step 1, we have 222 active elements and 945 active dofs.
Postprocessing: 
The relative error in QoI 0 is 4.9657720069292464e-05
The relative error in QoI 1 is 6.2108600947497457e-06

Using Adjoint Residual Error Estimator with Patch Recovery Weights
Refined mesh to 261 active elements and 1093 active dofs.
Assembling the System
Nonlinear Residual: 138340.35934180953
Linear solve starting, tolerance 9.9999999999999998e-13
Linear solve finished, step 64, residual 5.493304752738669e-15
Trying full Newton step
  Current Residual: 2.575339042156583e-05
  Nonlinear solver converged, step 0, residual reduction 1.8615963225839752e-10 < 9.9999999999999995e-08
  Nonlinear solver relative step size 0.00035395062927617756 > 9.9999999999999995e-08
Adaptive step 2, we have 261 active elements and 1093 active dofs.
Postprocessing: 
The relative error in QoI 0 is 3.248914175763547e-05
The relative error in QoI 1 is 9.154350541887689e-05

Using Adjoint Residual Error Estimator with Patch Recovery Weights
Refined mesh to 309 active elements and 1279 active dofs.
Assembling the System
Nonlinear Residual: 13524.651589128096
Linear solve starting, tolerance 9.9999999999999998e-13
Linear solve finished, step 73, residual 8.4716116062527396e-15
Trying full Newton step
  Current Residual: 2.9927786918380599e-05
  Nonlinear solver converged, step 0, residual reduction 2.2128323765795415e-09 < 9.9999999999999995e-08
  Nonlinear solver relative step size 0.00059108249457467238 > 9.9999999999999995e-08
Adaptive step 3, we have 309 active elements and 1279 active dofs.
Postprocessing: 
The relative error in QoI 0 is 1.3068055375400231e-05
The relative error in QoI 1 is 2.7384569868392317e-05

Using Adjoint Residual Error Estimator with Patch Recovery Weights
Refined mesh to 375 active elements and 1511 active dofs.
Assembling the System
Nonlinear Residual: 1654.7609046274454
Linear solve starting, tolerance 9.9999999999999998e-13
Linear solve finished, step 88, residual 5.351487781088356e-15
Trying full Newton step
  Current Residual: 2.8232088977040694e-05
  Nonlinear solver converged, step 0, residual reduction 1.706112883020819e-08 < 9.9999999999999995e-08
  Nonlinear solver relative step size 0.00035692025749874736 > 9.9999999999999995e-08
Adaptive step 4, we have 375 active elements and 1511 active dofs.
Postprocessing: 
The relative error in QoI 0 is 5.1850614729802354e-06
The relative error in QoI 1 is 5.8552193750982585e-07

Using Adjoint Residual Error Estimator with Patch Recovery Weights
Refined mesh to 438 active elements and 1737 active dofs.
Assembling the System
Nonlinear Residual: 6509.9123756624367
Linear solve starting, tolerance 9.9999999999999998e-13
Linear solve finished, step 101, residual 3.2722980646393311e-15
Trying full Newton step
  Current Residual: 2.7867536294505548e-05
  Nonlinear solver converged, step 0, residual reduction 4.2807851605943926e-09 < 9.9999999999999995e-08
  Nonlinear solver relative step size 0.00021329592182810594 > 9.9999999999999995e-08
Adaptive step 5, we have 438 active elements and 1737 active dofs.
Postprocessing: 
The relative error in QoI 0 is 1.5909681422576661e-06
The relative error in QoI 1 is 1.0292566222116434e-05

Using Adjoint Residual Error Estimator with Patch Recovery Weights
Refined mesh to 516 active elements and 2045 active dofs.
Assembling the System
Nonlinear Residual: 5371.8919704253385
Linear solve starting, tolerance 9.9999999999999998e-13
Linear solve finished, step 107, residual 2.3500869008725603e-15
Trying full Newton step
  Current Residual: 2.6527045411426849e-05
  Nonlinear solver converged, step 0, residual reduction 4.938119671331826e-09 < 9.9999999999999995e-08
  Nonlinear solver relative step size 0.00012064223575858791 > 9.9999999999999995e-08
Adaptive step 6, we have 516 active elements and 2045 active dofs.
Postprocessing: 
The relative error in QoI 0 is 3.7224084862796006e-07
The relative error in QoI 1 is 7.9879129730399771e-06


 -------------------------------------------------------------------------------------------------------------------
| Time:           Wed Apr 13 10:20:38 2016                                                                          |
| OS:             Darwin                                                                                            |
| HostName:       peterson-laptop.local                                                                             |
| OS Release:     15.0.0                                                                                            |
| OS Version:     Darwin Kernel Version 15.0.0: Sat Sep 19 15:53:46 PDT 2015; root:xnu-3247.10.11~1/RELEASE_X86_64  |
| Machine:        x86_64                                                                                            |
| Username:       petejw                                                                                            |
| Configuration:  ../configure  'INSTALL=/usr/bin/install -C'                                                       |
|  '--with-methods=opt dbg'                                                                                         |
|  '--prefix=/Users/petejw/projects/libmesh_git/installed'                                                          |
|  '--enable-perflog'                                                                                               |
|  '--enable-tracefiles'                                                                                            |
|  '--disable-blocked-storage'                                                                                      |
|  '--enable-default-comm-world'                                                                                    |
|  '--enable-tecio'                                                                                                 |
|  '--enable-unique-ptr'                                                                                            |
|  '--enable-ifem'                                                                                                  |
|  'PETSC_DIR=/opt/moose/petsc/mpich_petsc-3.6.1/clang-opt-superlu'                                                 |
|  'SLEPC_DIR=/opt/petejw/slepc/3.6.1'                                                                              |
|  'CPPFLAGS= '                                                                                                     |
|  'LIBS= '                                                                                                         |
|  'CPPFLAGS=  '                                                                                                    |
|  'LIBS=  '                                                                                                        |
|  '--disable-netcdf-4'                                                                                             |
|  '--disable-testsets'                                                                                             |
|  'CXX=mpicxx'                                                                                                     |
|  'CC=mpicc'                                                                                                       |
|  'F77=mpif77'                                                                                                     |
|  'FC=mpif90'                                                                                                      |
|  'CPPFLAGS=   '                                                                                                   |
|  'LIBS=   '                                                                                                       |
 -------------------------------------------------------------------------------------------------------------------
 ----------------------------------------------------------------------------------------------------------------------
| libMesh Performance: Alive time=0.905286, Active time=0.864229                                                       |
 ----------------------------------------------------------------------------------------------------------------------
| Event                                   nCalls     Total Time  Avg Time    Total Time  Avg Time    % of Active Time  |
|                                                    w/o Sub     w/o Sub     With Sub    With Sub    w/o S    With S   |
|----------------------------------------------------------------------------------------------------------------------|
|                                                                                                                      |
|                                                                                                                      |
| AdjointResidualErrorEstimator                                                                                        |
|   estimate_error()                      6          0.0412      0.006864    0.2087      0.034777    4.77     24.14    |
|                                                                                                                      |
| DofMap                                                                                                               |
|   add_neighbors_to_send_list()          19         0.0014      0.000073    0.0014      0.000073    0.16     0.16     |
|   build_constraint_matrix()             12726      0.0112      0.000001    0.0112      0.000001    1.29     1.29     |
|   build_sparsity()                      7          0.0073      0.001038    0.0080      0.001143    0.84     0.93     |
|   cnstrn_elem_mat_vec()                 2121       0.0018      0.000001    0.0018      0.000001    0.21     0.21     |
|   constrain_elem_matrix()               2121       0.0018      0.000001    0.0018      0.000001    0.21     0.21     |
|   constrain_elem_vector()               6363       0.0022      0.000000    0.0022      0.000000    0.26     0.26     |
|   create_dof_constraints()              19         0.0020      0.000104    0.0027      0.000140    0.23     0.31     |
|   distribute_dofs()                     19         0.0017      0.000088    0.0090      0.000472    0.19     1.04     |
|   dof_indices()                         71459      0.0164      0.000000    0.0164      0.000000    1.90     1.90     |
|   enforce_adjoint_constraints_exactly() 24         0.0014      0.000059    0.0014      0.000059    0.16     0.16     |
|   enforce_constraints_exactly()         24         0.0012      0.000048    0.0012      0.000048    0.13     0.13     |
|   old_dof_indices()                     19089      0.0054      0.000000    0.0054      0.000000    0.62     0.62     |
|   prepare_send_list()                   32         0.0000      0.000001    0.0000      0.000001    0.00     0.00     |
|   reinit()                              19         0.0073      0.000384    0.0073      0.000384    0.85     0.85     |
|                                                                                                                      |
| EquationSystems                                                                                                      |
|   build_discontinuous_solution_vector() 12         0.0008      0.000067    0.0015      0.000125    0.09     0.17     |
|   build_solution_vector()               21         0.0064      0.000306    0.0088      0.000418    0.74     1.02     |
|                                                                                                                      |
| FE                                                                                                                   |
|   compute_shape_functions()             38196      0.0395      0.000001    0.0395      0.000001    4.57     4.57     |
|   init_shape_functions()                9917       0.0643      0.000006    0.0643      0.000006    7.44     7.44     |
|   inverse_map()                         21880      0.0154      0.000001    0.0154      0.000001    1.78     1.78     |
|                                                                                                                      |
| FEMSystem                                                                                                            |
|   assemble_qoi_derivative()             7          0.0137      0.001956    0.0369      0.005266    1.58     4.27     |
|   assembly()                            7          0.0118      0.001688    0.0529      0.007555    1.37     6.12     |
|   assembly(get_jacobian)                7          0.0127      0.001811    0.0542      0.007738    1.47     6.27     |
|   assembly(get_residual)                7          0.0075      0.001070    0.0244      0.003481    0.87     2.82     |
|   numerical_elem_jacobian()             4626       0.0379      0.000008    0.0379      0.000008    4.39     4.39     |
|   numerical_side_jacobian()             1344       0.0073      0.000005    0.0073      0.000005    0.85     0.85     |
|   postprocess()                         7          0.0057      0.000812    0.0225      0.003219    0.66     2.61     |
|                                                                                                                      |
| FEMap                                                                                                                |
|   compute_affine_map()                  38196      0.0323      0.000001    0.0323      0.000001    3.74     3.74     |
|   compute_face_map()                    3360       0.0051      0.000002    0.0133      0.000004    0.59     1.54     |
|   init_face_shape_functions()           35         0.0000      0.000001    0.0000      0.000001    0.00     0.00     |
|   init_reference_to_physical_map()      9917       0.0355      0.000004    0.0355      0.000004    4.11     4.11     |
|                                                                                                                      |
| GMVIO                                                                                                                |
|   write_nodal_data()                    21         0.1002      0.004771    0.1002      0.004771    11.59    11.59    |
|                                                                                                                      |
| GenericProjector                                                                                                     |
|   copy_dofs                             5067       0.0026      0.000001    0.0117      0.000002    0.30     1.35     |
|   operator()                            18         0.0054      0.000299    0.2448      0.013600    0.62     28.32    |
|   project_edges                         1296       0.0001      0.000000    0.0001      0.000000    0.01     0.01     |
|   project_interior                      1296       0.0001      0.000000    0.0001      0.000000    0.01     0.01     |
|   project_nodes                         1296       0.0046      0.000004    0.2228      0.000172    0.54     25.78    |
|   project_sides                         1296       0.0001      0.000000    0.0001      0.000000    0.01     0.01     |
|                                                                                                                      |
| ImplicitSystem                                                                                                       |
|   adjoint_solve()                       7          0.0002      0.000031    0.1304      0.018635    0.03     15.09    |
|                                                                                                                      |
| Mesh                                                                                                                 |
|   all_first_order()                     12         0.0025      0.000209    0.0025      0.000209    0.29     0.29     |
|   all_second_order()                    1          0.0000      0.000021    0.0000      0.000021    0.00     0.00     |
|   contract()                            6          0.0007      0.000109    0.0024      0.000399    0.08     0.28     |
|   find_neighbors()                      33         0.0226      0.000686    0.0226      0.000686    2.62     2.62     |
|   renumber_nodes_and_elem()             30         0.0019      0.000064    0.0019      0.000064    0.22     0.22     |
|                                                                                                                      |
| MeshOutput                                                                                                           |
|   write_equation_systems()              21         0.0002      0.000009    0.1092      0.005199    0.02     12.63    |
|                                                                                                                      |
| MeshRefinement                                                                                                       |
|   _coarsen_elements()                   12         0.0001      0.000006    0.0001      0.000006    0.01     0.01     |
|   _refine_elements()                    15         0.0033      0.000221    0.0055      0.000365    0.38     0.63     |
|   add_node()                            6156       0.0016      0.000000    0.0016      0.000000    0.19     0.19     |
|   make_coarsening_compatible()          17         0.0025      0.000145    0.0025      0.000145    0.29     0.29     |
|   make_flags_parallel_consistent()      18         0.0004      0.000024    0.0004      0.000024    0.05     0.05     |
|   make_refinement_compatible()          17         0.0002      0.000014    0.0002      0.000014    0.03     0.03     |
|                                                                                                                      |
| NewtonSolver                                                                                                         |
|   solve()                               7          0.0009      0.000130    0.1199      0.017130    0.11     13.88    |
|                                                                                                                      |
| OldSolutionValue                                                                                                     |
|   Number eval_at_node()                 11664      0.0008      0.000000    0.0008      0.000000    0.10     0.10     |
|   check_old_context(c)                  5067       0.0032      0.000001    0.0068      0.000001    0.37     0.79     |
|   check_old_context(c,p)                6060       0.0034      0.000001    0.0084      0.000001    0.40     0.97     |
|   eval_at_point()                       6060       0.1069      0.000018    0.2159      0.000036    12.37    24.98    |
|   eval_old_dofs()                       5067       0.0014      0.000000    0.0086      0.000002    0.16     1.00     |
|                                                                                                                      |
| Parallel                                                                                                             |
|   allgather()                           19         0.0000      0.000000    0.0000      0.000000    0.00     0.00     |
|                                                                                                                      |
| Partitioner                                                                                                          |
|   set_node_processor_ids()              24         0.0014      0.000059    0.0014      0.000059    0.16     0.16     |
|   single_partition()                    21         0.0003      0.000016    0.0003      0.000016    0.04     0.04     |
|                                                                                                                      |
| PatchRecoveryErrorEstimator                                                                                          |
|   estimate_error()                      18         0.1024      0.005690    0.1400      0.007776    11.85    16.20    |
|                                                                                                                      |
| PetscLinearSolver                                                                                                    |
|   solve()                               21         0.0792      0.003772    0.0792      0.003772    9.17     9.17     |
|                                                                                                                      |
| System                                                                                                               |
|   project_vector(old,new)               18         0.0043      0.000240    0.2523      0.014014    0.50     29.19    |
|                                                                                                                      |
| TopologyMap                                                                                                          |
|   init()                                15         0.0120      0.000799    0.0120      0.000799    1.39     1.39     |
 ----------------------------------------------------------------------------------------------------------------------
| Totals:                                 292283     0.8642                                          100.00            |
 ----------------------------------------------------------------------------------------------------------------------

 
***************************************************************
* Done Running Example adjoints_ex1:
*   example-opt  
***************************************************************
</pre>
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